What is an activity vector in capsule neural networks?
In a traditional neural network, the network's vertices are neurons and the output of a single neuron is a single value (a &quot;scalar&quot;). This number is called its activation. A layer of neurons in the network outputs a vector of activations. We should not confuse this with the activity vectors in a Capsule Network.
Capsule Networks are different since the network vertices are Capsules rather than neurons. They are a higher-dimensional: the output of a Capsule is not a scalar but a vector representing a group of parameters related to the input. Hence the name activation vector.
Motivation
In a neural network, there is no inherent structure between the scalar outputs of the neurons, this is something the following layers have to learn. In Capsule Networks the output of a capsule represents all the parameters related to that together in a vector including a prediction for the activation of deeper layer Capsules. This adds a useful local structure.
For example, consider face recognition. If you have a capsule that knows how to recognize eyes it could output an activity vector representing e.g. &quot;since I have recognized an eye position $(x,y)$ with probability $p=0.97$ I predict the parameters for the whole face will be $(f_1,\dots, f_n)$&quot;.
As explained in the Dynamic Routing Between Capsules paper you refer to this information is then used in the way that the capsules in earlier layers (the parts: eye, mouth, nose) predict the activations of deeper layers (face). For example, a face recognizer will only be strongly activated when there is an agreement between the eye, nose and mouth recognizers (the parts) and the face recognizer (the whole) about where the face is located (the $(f_1,\dots, f_n)$ parameters).
Historical Inspiration
Older computer vision algorithms like SIFT work in a similar way where recognition is based on agreement between the configuration of multi-dimensional features (key points) and the reference configuration.